A discrete learning method for nonlinear problems involving extreme deformation

Nonlinear problems with extreme deformation are important in many engineering fields and how to develop the interpretable machine learning method for them is still a critical and open question. As a non-parametric and interpretable Bayesian model, Gaussian process regression (GPR) is promising for real-time prediction but suffers from the curse of dimensionality in problems with large sample size and is time-consuming in predicting high-dimensional problems. In this study, a discrete machine learning method (DLM) is proposed to alleviate the curse of dimensionality of GPR and extend it to large-scale nonlinear problems with extreme deformation. In this method, for a given training dataset, a dual-discretization algorithm is introduced to divide it into multiple subsets that are suitable for the GPR. The dual-discretization algorithm consists of two steps: solution space discretization and material domain discretization, which first clusters the training points into multiple subspaces according to structural responses and then divides the material domain into multiple subdomains based on the degree of deformation. Moreover, a selective strategy is proposed for hyperparameter optimizations of local GPRs associated with subdomains within the same subspace. A high-fidelity numerical simulator is applied, using the B-spline material point method, to generate training data for a set of extreme deformation problems with increasing dimension sizes for demonstration. It is shown that the proposed DLM can take less than 1 minute to complete the online predictions with desirable accuracy for large-scale nonlinear extreme deformation problems with over 10 million degrees of freedom,which is not achievable by the traditional GPR. The proposed DLM is expected to serve as a powerful tool for the fast and accurate prediction of large-scale nonlinear problems involving extreme deformation.